Normalized defining polynomial
\( x^{18} - x^{15} - 8x^{12} + 17x^{9} - 8x^{6} - x^{3} + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(43318105769609220096\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 29^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(12.33\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2\cdot 3^{1/2}29^{1/2}\approx 18.65475810617763$ | ||
Ramified primes: | \(2\), \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{6}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{6}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{6}a^{14}-\frac{1}{2}a^{8}+\frac{1}{3}a^{7}-\frac{1}{2}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{15}-\frac{1}{2}a^{9}+\frac{1}{3}a^{8}-\frac{1}{2}a^{3}-\frac{1}{3}a$, $\frac{1}{6}a^{16}-\frac{1}{2}a^{10}+\frac{1}{3}a^{9}-\frac{1}{2}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{6}a^{17}-\frac{1}{2}a^{11}+\frac{1}{3}a^{10}-\frac{1}{2}a^{5}-\frac{1}{3}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a$, $\frac{1}{6}a^{17}-\frac{5}{6}a^{16}+\frac{1}{6}a^{15}-\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-\frac{1}{6}a^{12}-\frac{7}{6}a^{11}+\frac{41}{6}a^{10}-\frac{3}{2}a^{9}+\frac{11}{2}a^{8}-\frac{23}{2}a^{7}+\frac{17}{6}a^{6}-\frac{11}{2}a^{5}+\frac{13}{6}a^{4}-\frac{1}{6}a^{3}+\frac{7}{6}a^{2}+\frac{3}{2}a-\frac{7}{6}$, $\frac{1}{3}a^{17}+\frac{1}{6}a^{15}-\frac{1}{3}a^{13}-\frac{8}{3}a^{11}-\frac{3}{2}a^{9}+3a^{8}+\frac{8}{3}a^{7}+\frac{4}{3}a^{6}+\frac{1}{3}a^{5}-3a^{4}+\frac{3}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{2}{3}$, $\frac{3}{2}a^{17}+\frac{1}{3}a^{16}-\frac{1}{6}a^{15}-\frac{1}{2}a^{14}-\frac{1}{3}a^{13}+\frac{1}{6}a^{12}-\frac{25}{2}a^{11}-\frac{8}{3}a^{10}+\frac{3}{2}a^{9}+\frac{103}{6}a^{8}+\frac{17}{3}a^{7}-\frac{5}{2}a^{6}+\frac{5}{6}a^{5}-\frac{8}{3}a^{4}+\frac{1}{2}a^{3}-\frac{5}{2}a^{2}+\frac{1}{2}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{16}-\frac{5}{6}a^{15}-\frac{1}{6}a^{13}+\frac{1}{3}a^{12}-\frac{3}{2}a^{11}+\frac{3}{2}a^{10}+\frac{41}{6}a^{9}+\frac{4}{3}a^{8}-\frac{1}{6}a^{7}-10a^{6}+\frac{7}{6}a^{5}-\frac{17}{6}a^{4}+\frac{7}{6}a^{3}-\frac{1}{3}a^{2}+\frac{3}{2}a+1$, $\frac{1}{6}a^{17}-\frac{2}{3}a^{16}+\frac{1}{6}a^{15}-\frac{1}{6}a^{14}+\frac{1}{6}a^{13}-\frac{1}{3}a^{12}-\frac{3}{2}a^{11}+\frac{17}{3}a^{10}-\frac{3}{2}a^{9}+\frac{17}{6}a^{8}-\frac{43}{6}a^{7}+4a^{6}-\frac{1}{6}a^{5}-\frac{5}{3}a^{4}-\frac{11}{6}a^{3}-\frac{3}{2}a^{2}+\frac{11}{6}a-\frac{2}{3}$, $\frac{1}{3}a^{17}-\frac{5}{6}a^{16}+a^{15}-\frac{1}{6}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{8}{3}a^{11}+\frac{41}{6}a^{10}-\frac{25}{3}a^{9}+\frac{25}{6}a^{8}-10a^{7}+\frac{34}{3}a^{6}-\frac{4}{3}a^{5}+\frac{5}{6}a^{4}+\frac{1}{2}a^{2}+\frac{2}{3}a-\frac{4}{3}$, $\frac{1}{3}a^{17}-\frac{3}{2}a^{16}+\frac{1}{6}a^{15}-\frac{1}{6}a^{14}+\frac{2}{3}a^{13}-\frac{8}{3}a^{11}+\frac{25}{2}a^{10}-\frac{3}{2}a^{9}+\frac{9}{2}a^{8}-\frac{56}{3}a^{7}+\frac{4}{3}a^{6}-\frac{2}{3}a^{5}+\frac{1}{2}a^{4}+\frac{3}{2}a^{3}-\frac{5}{6}a^{2}+\frac{11}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{16}+\frac{1}{3}a^{15}-\frac{1}{6}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{7}{6}a^{11}+\frac{3}{2}a^{10}-3a^{9}+\frac{17}{6}a^{8}-4a^{7}+\frac{16}{3}a^{6}-\frac{17}{6}a^{5}+\frac{11}{6}a^{4}-\frac{2}{3}a^{3}+\frac{7}{6}a^{2}+a-\frac{4}{3}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 230.414214587 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 230.414214587 \cdot 1}{2\cdot\sqrt{43318105769609220096}}\cr\approx \mathstrut & 0.170076175640 \end{aligned}\]
Galois group
$S_3\times D_6$ (as 18T29):
A solvable group of order 72 |
The 18 conjugacy class representatives for $S_3\times D_6$ |
Character table for $S_3\times D_6$ |
Intermediate fields
\(\Q(\sqrt{29}) \), 3.1.87.1, 3.1.116.1, 6.2.219501.1, 6.2.390224.1, 9.1.1222181568.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 12.0.2111929749504.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
\(3\) | 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |